A Little Theory

In the first part of this article, I present some statistical theory to describe leveraged ETFs and various two-fund portfolios. If you don't have the statistical background or just aren't interested in this part, you can skip to the "Illustration for 2x Multiple" section and go from there.

How Leveraged ETFs Amplify Returns and Volatility

Most leveraged ETFs aim to multiply the daily gains of some underlying index by a constant. For example, the ProShares UltraPro S&P 500 ETF (NYSEARCA:UPRO ) aims to multiply daily gains of the S&P 500 by a factor of 3. In recent articles, I showed that UPRO does an excellent job tracking 3x the index.

Daily returns of UPRO are well described by the simple linear regression model:

Y = alpha + beta X + e, e

(0, σ 2 )

This just says that UPRO's daily gains, Y, are linearly related to the S&P 500's daily gains, X. The e term indicates that the relationship isn't quite perfect, i.e. there are random fluctuations around the regression line. We assume that these fluctuations have mean 0 and some constant variance σ 2 .

If UPRO operated perfectly, alpha would be 0, beta would be 3, and σ 2 would be 0. In other words, daily gains would be exactly 3x that of the S&P 500.

In practice, this isn't far off. In the 1,445 trading days from UPRO's inception on June 25, 2009 to March 23, 2015, a fitted regression model estimated alpha to be 0.0191, beta to be 2.9535, and σ 2 to be 0.0629. Therefore UPRO approximately follows the Y = 3X line. This is clear in the scatterplot shown below, where it's hard to even distinguish the blue regression line from the red Y = 3X line.

Expected Return and Volatility

Using basic statistical properties, the expected daily return of UPRO is E[Y] = E[alpha + beta X + e] = alpha + beta E[X], or approximately 3 E[X]. So the expected daily return of UPRO is 3x the expected daily return of the S&P 500, as you would expect.

The variance of daily returns for UPRO is given by V[Y] = V[alpha + beta X + e) = beta 2 V[X] + σ 2. This should be approximately 9 V[X] for UPRO, since σ 2 is very small and beta is approximately 3. This means that the standard deviation of daily returns for UPRO is approximately 3 SD[X], or three times the standard deviation of daily returns for the S&P 500. So the volatility of UPRO is 3x the volatility of the S&P 500.

In practice, we could estimate the exact expected return of UPRO as 0.0191 + 2.9535 E[X], and its variance as 0.0629 + 2.9535 2 V[X] = 0.0629 + 8.723 V[X]. We could plug in the S&P 500's historical mean since 1950 (0.0342%) for E[X] and its historical variance (0.9374%) for V[X] to get an estimated daily return for UPRO of 0.1201% and an estimate variance of 8.2399%. This would correspond to a return-volatility ratio of 0.0418, which, unexpectedly, is a bit higher than the S&P 500's ratio of 0.0353.

Where Are UPRO's Expenses?

The fact that UPRO appears to have a better expected return-volatility ratio than the S&P 500 to date is probably a random thing. Theoretically, leveraged ETFs should always have a slightly lower return-volatility ratio than the underlying index. The reason it was a bit higher in UPRO's case was that the positive alpha dominated the small σ 2. To clarify, the positive alpha inflated the 3x excess return, and the positive σ 2 inflated the 3x extra volatility. But the effect of alpha was stronger.

Why on earth was alpha positive to begin with? I would expect UPRO's 0.95% expense ratio to manifest itself in a slightly negative alpha when regressing daily gains of UPRO vs. the S&P 500. But for some reason that hasn't been the case over UPRO's short lifetime. Maybe the bull market has been such that UPRO's operating expenses have been less than expected, but I would think that ProShares would pocket the 0.95% regardless. And, anyway, even with a 0% expense ratio you wouldn't expect a positive alpha.

Any thoughts on where UPRO's fees show up in its performance? Share your thoughts in the comments section.

Two-Fund Portfolios With UPRO

It's pretty cool that we can use UPRO to amplify expected returns while maintaining the same expected return-volatility ratio as the S&P 500. It would be even cooler if we could amplify returns while increasing the return-volatility ratio. But that's a story for another day.

I want to consider a few potential two-fund portfolios consisting of UPRO and some other investment. I should start by defining my notation.

Let Z represent daily gains of a two-fund portfolio, consisting of a weighted combination of UPRO and some other investment. If Y1 represents daily gains for UPRO, and Y2 represent daily gains for the other fund, we can assume the following models:

Then daily gains for our portfolio can be written as:

where c1 is the proportion of our money in UPRO, and c2 is the proportion in the other fund. It is not too hard to work out the expected value and variance of our daily portfolio gain, Z:

These formulas simplify somewhat when we consider the best-case scenario where UPRO acts as a perfect 3x ETF with no fees, i.e. Y1 = 3X. Then we get:

So the expected value of Z can be greater or less than 3 E[X], and the standard deviation can be greater than or equal to 3 SD[X], depending on the weights and the parameters governing the second fund's regression on daily S&P 500 gains.

UPRO + Cash = Aggression Tuning

The expected return-volatility ratio becomes E[X]/SD[X], just like it was for the S&P 500 and UPRO. But we can choose c1 to control the raw expected return. In that sense, UPRO can serve as a sort of aggression adjustment tuner, where you can choose your allocation in UPRO to achieve any target multiple of the S&P 500 between 0 and 3x. The expected return-volatility ratio is the same regardless of the allocation.

To achieve some target multiple d with a UPRO/cash combination, we need to put (d/3 x 100)% of our funds in UPRO, and the remaining percentage in cash.

If we really want 1.25x the S&P 500's expected return (and volatility), we can either invest in an available 1.25x ETF like the Direxion Daily S&P 500 Bull 1.25x Shares (NYSEARCA:LLSP ), or invest 41.7% of our money in UPRO and hold the other 58.3% in cash. The effective expense ratio in the UPRO/cash combination would be 0.40%, slightly lower than the net expense ratio for LLSP, which is 0.50%.

An interesting special case is where you put one-third of your money in UPRO and two-thirds in cash. At the onset, this portfolio would behave almost exactly as if you had all of your money in the S&P 500. UPRO's expense ratio should result in somewhat diminished returns, but not much. And it might be worth it to free up two-thirds of your money, for emergencies and so forth.

Rebalancing Costs

An important consideration before attempting a two-fund portfolio with UPRO is the taxes and transaction fees that you might have to deal with. Let's start with capital gains taxes. Consider the one-third UPRO, two-thirds cash idea. Suppose you start with $1,000 in cash and $2,000 in UPRO, and don't rebalance over the first year. If the S&P 500 gains 10% and UPRO gains 30% (I know, UPRO won't necessarily achieve 3x annual growth), then both the 100% S&P 500 portfolio and the UPRO/cash portfolio would have gained $300. You would owe the same long-term capital gains taxes that would be equivalent for either case.

However, suppose you had to rebalance a few times within the year, selling some shares of

UPRO to convert to cash. You would owe short-term capital gains taxes on those earnings. The 100% S&P 500 portfolio doesn't have that problem. Of course, if you're operating within a Roth IRA, you're free to buy and sell within the account as much as you like without paying capital gains taxes.

As for transaction fees, note that a 3x leveraged ETF can be extremely volatile. In a good month, your holdings in UPRO might increase by 15% or more. This would require rebalancing to get back to the one-third UPRO, two-thirds cash allocation. That means selling some shares of UPRO and paying the $7 (or whatever you pay) transaction fee.

How damaging can transaction fees be? Well, that depends on how tightly you want to keep the proportion near one-third, two-thirds. The more willing you are to let the weights deviate from that allocation, the less you have to rebalance (and the less your portfolio behaves like the S&P 500). It also depends on the value of your portfolio. Transaction fees are obviously more damaging to a $3,000 portfolio than a $100,000 portfolio.

If our second fund is an S&P 500 index fund, we have E[Y2 ] = E[X]. Then E[Z] = 3c1 E[X] + c2 E[X], which simplifies to (2c1 + 1) E[X]. We can achieve any multiple between 1 and 3 with this combination. For a target daily multiple d, we need to put [(d-1)/2 x 100]% of our money in UPRO and the remaining percentage in the S&P 500.

The advantage of the UPRO/S&P 500 portfolio compared to UPRO/cash is that you won't have to rebalance as often, since the two funds move in the same direction. The disadvantage is that each rebalancing costs you twice as much in transaction fees, since you have to execute two trades rather than one. Other disadvantages are that you're not holding on to cash, and you're paying a (probably small) expense ratio on the S&P 500 holdings, which you're not paying in the UPRO/cash portfolio.

If our second fund was a leveraged ETF with a different multiple, we'd have a very similar scenario. To achieve a target multiple d with UPRO and a b-times S&P 500 ETF, you would need to put [(d-1)/(3-b) x 100]% in UPRO and the remaining amount in the other leveraged ETF. You could achieve any daily multiple between b and 3. Again, you would have to make two trades each time you rebalance, but you would have to rebalance even less frequently than for the UPRO/S&P 500 portfolio. You would probably pay a substantial expense ratio on the "other ETF" holdings, much higher than the S&P 500 index fund.

Another consideration is that the volatility of a UPRO/other leveraged ETF portfolio might be inflated slightly due to tracking error of both funds. Following the statistical notation from earlier, both c 1 σ1 2 and c2 2 σ2 2 would contribute to V[Z] rather than just c1 2 σ1 2. My sense is the extra contribution to V[Z] would still be negligible, but it could theoretically inflate variance and reduce the expected return-volatility ratio of the portfolio a bit.

Illustration for 2x Multiple

Let's say we had $10,000 five years ago and wanted to put it to work to achieve a 2x daily multiple of the S&P 500. Consider three different implementations:

Our target multiple is 2x, so let's rebalance the UPRO/cash and UPRO/SPY portfolios whenever our allocations are such that we get below 1.9x or above 2.1x. That translates to the UPRO allocation falling outside of 63.3%-70.0% for the UPRO/cash portfolio, and outside of 45.0%-55.0% for the UPRO/SPY portfolio. We'll pay a $7 fee each time we rebalance UPRO/cash and a $14 fee each time we rebalance UPRO/SPY.

Here are the results.

Final balances were $31,228.91 for UPRO/cash, $32,649.06 for UPRO/SPY, and $32,506.66 for SSO.

Strangely, all of these were greater than the final balance for a hypothetical "perfect" 2x S&P 500 ETF, which was $28.255.21. This is probably related to UPRO and SSO's positive alphas over the last 5 years. For some reason, these funds have been doing better than they really should be doing. Anyway, the point is that these three approaches operate very similarly.

Note that we had to rebalance the UPRO/cash portfolio 31 times and the UPRO/SPY portfolio 10 times over this 5-year period. This translates to $217 in transaction fees for UPRO/cash (2.2% of the initial $10,000 balance) and $140 in transaction fees for UPRO/SPY (1.4% of the initial $10,000 balance). If we were a bit more flexible and allowed the effective multiple to go from 1.8-2.2 rather than 1.9-2.1, we would have only rebalanced UPRO/cash 10 times, and UPRO/SPY 2 times, bringing down the percentages to 0.7% and 0.28% of the initial $10,000 balances.

Closing Comments

The main message of this article is that you can combine leveraged ETFs with cash or with each other to achieve any target daily multiple of an index that you want (up to the highest multiple of the available ETFs). If you're dealing with leveraged ETFs that track their index very well, then any allocation of leveraged ETFs and cash will have the same expected return-volatility ratio as the underlying index. So you can basically use leveraged ETFs to achieve some desired level of aggression, without lowering your risk-adjusted return.

You will have to execute one trade each time you rebalance for an ETF/cash portfolio, and two trades each time you rebalance for a two-ETF portfolio. The more flexible you are in allowing the target allocation to deviate a bit from the target, the less often you'll have to rebalance, and the lower these costs will be. And the greater the balance in your portfolio, the less damaging these fees become. For example, if you have a $50,000 portfolio and use SSO/cash to achieve a 1.5x daily multiple of the S&P 500, even if you have a volatile year where you have to rebalance once every month, your transaction fees would only represent 0.17% of your initial balance (assuming $7 trades).

Outside of a Roth IRA, rebalancing in less than one-year intervals means owing short-term capital gains tax on earnings, which could render these two-fund strategies imprudent. One solution for non-Roth IRA investors would be to rebalance only once per year, thus subjecting earnings to the normal long-term capital gains taxes. You would have to be open to the prospect of your portfolio becoming overly aggressive during the course of a strong year. On the flipside, when the market's down and you're losing money, you can rebalance without worrying about capital gains taxes, because you're realizing losses rather than gains.

For those aware of and tolerant of the costs, implementing one of these two-fund strategies has some nice advantages. One, it can free up a lot of cash. A one-third UPRO, two-thirds cash portfolio behaves almost exactly like a 100% S&P 500 portfolio, except you get to hold on to a lot of cash. Well, and you probably pay a slightly higher effective expense ratio ((1/3)(0.95%) = 0.32%) compared to an S&P 500 index fund, and have to rebalance occasionally.

Another advantage is that in some cases you might be able to replicate the target daily multiple of an available leveraged ETF, while slightly reducing your effective expense ratio. For example, you can achieve a 2x S&P 500 daily multiple with one-half UPRO, one-half SPY portfolio. This combination would have an effective expense ratio of (1/2)(0.95%) + (1/2)(0.09%) = 0.52%. SSO achieves the same 2x daily multiple, but with a 0.85% expense ratio.

You can do a lot with leveraged ETFs, particularly if you want any of the following:

To achieve a particular target daily multiple of the index not offered by available leveraged ETFs.

To achieve a target daily multiple offered by an available leveraged ETF, but with a slightly lower effective expense ratio.

To put your money to work while having access to cash.

Disclosure: The author is long UPRO. (More. ) The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.

Additional disclosure: The author used Yahoo! Finance to obtain historical prices for the S&P 500, UPRO, SSO, and SPY, and used R to analyze the data and generate figures. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.